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Chapter 11: Q. 6 (page 901)

Finding limits: Find the given limits if they exist. If a limit does not exist, explain why.

limt鈫鈭3鈥|t+3|t+3,|t+3|t鈭3

Short Answer

Expert verified

limt鈫鈭3鈥|t+3|t+3,|t+3|t鈭3=1,0

Step by step solution

01

Step 1. Given information

limt鈫鈭3鈥|t+3|t+3,|t+3|t鈭3

02

Step 2. Finding limits for limt→−3 |t+3|t+3

If鈥limx鈫a-fx鈮limx鈫a+fx鈥then鈥the鈥limit鈥does鈥not鈥exist

limx鈫鈭3+鈥|x+3|x+3=1

limx鈫鈭3鈥鈭|x+3|x+3=鈭1

03

Step 3. Finding limit for limt→−3 |t+3|t-3

plug in the value

limx鈫鈭3鈥|x+3|x鈭3=|鈭3+3|鈭3鈭3

limx鈫鈭3鈥|x+3|x鈭3=0

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Most popular questions from this chapter

Every description of the DNA molecule says that the strands of the helices run in opposite directions. This is meant as a statement about chemistry, not about the geometric shape of the double helix. Consider two helices

h1(t)=鉄cos鈦t,sin鈦t,伪t鉄andh2(t)=鉄sin鈦t,cos鈦t,伪t鉄

(a) Sketch these two helices in the same coordinate system, and show that they run geometrically in different directions.

(b) Explain why it is impossible for these two helices to fail to intersect, and hence why they could not form a configuration for DNA.

Find and graph the vector function determined r(t)=鉄x(t),y(t)鉄by the differential equation

x鈥(t)=x,y鈥(t)=x2,x(0)=1,y(0)=2. (HINT: Start by solving the initial-value problem x鈥(t)=x,x(0)=1.)

Find parametric equations for each of the vector-valued functions in Exercises 26鈥34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

r(t)=(tsint,tcost,t)fort鈭[0,4蟺]

Find parametric equations for each of the vector-valued functions in Exercises 26鈥34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

r(t)=(1+sint,3-cos2t),fort鈭[0,2蟺]

Find and graph the vector function determined r(t)=鉄x(t),y(t)鉄by the differential equation

x鈥(t)=鈭y,y鈥(t)=x,x(0)=1,y(0)=0. ( HINT: What familiar pair of functions have the given properties ?)
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